Add 1½ cups chopped peeled ginger to a saucepan filled with 2 cups water and bring to a low simmer; cook, partially covered, 45 minutes. Remove from heat and let steep, covered, 20 minutes.

Pour mixture through a fine mesh strainer into a bowl, pressing on ginger pieces and then discarding. Return liquid to saucepan and add ¾ cup sugar and a pinch of kosher salt, then heat over medium heat, stirring, until sugar has dissolved. Chill syrup in a covered jar until cold. Do Ahead: Ginger syrup can be made 1 week ahead; store chilled in an airtight container (such as a mason jar).

Pour ¼ cup ginger syrup and 1½ tsp. lime juice into a glass filled with ice and stir to combine. Add ¾ cup chilled seltzer or club soda and stir gently. Taste and adjust with more syrup, seltzer, or lime juice as desired, and garnish with a lime wedge.